The latent factors are known as state variables in the jargon. Naturally, it would be desirable to have the associated error bounds for the nowcast. They do not, however, exploit the factor time series dynamics imposed by \({\mathbf {B}}(L)\), or the fact that the idiosyncratic components are potentially cross-sectionally heteroscedastic (i. (2015d, Sect.
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1007/BF02294246Instant access to the full article PDF. Explore systematic policy: A change in regimeNext, we contemplate a shift in policy regime. , Bernanke et al. If the state space model is not Gaussian, then the Kalman filter and smoother do not in general provide the conditional means, and the associated estimators are no longer the minimum MSE estimators. , Hamilton (1994, p. In essence, as \(N \rightarrow \infty \), all eigenvalues of \(N^{-1}{\varvec{{\Lambda }}}'{\varvec{{\Lambda }}}\) should be positive and finite, implying that the \({\mathcal {R}}\) largest eigenvalues of \({\varvec{{\Sigma }}}\) are unbounded asymptotically.
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white noise with limited cross-sectional dependence: \(E(u_{i,t})=0\), \(E(u_{i,t}u_{j,s}) = 0\) for all \(t\ne s\), and \(E(u_{i,t}u_{j,t}) = \tau _{i,j}\), with \(\sum _{i=0}^N |\tau _{i,j}| {\mathcal {J}}\), where \({\mathcal {J}}\) is some positive number that does not depend on N or T. I generate the variable ip as the annualized quarterly growth rate of industrial production and use a sample from 1954 to 2006. Radaelli, Springer Lecture Notes in Computational and Applied Mathematics 23, pp. An equation for one of the variables X~1~, \[eq\_eq0\] \[eq\_eq1\] \[eq\_eq2\] \[eq\_eq3\] \[eq\_eq4\] The analysis begins with the dimension 1. We can obtain the one-step predictions of the factor by typing
We can then trace the path of the factor by graphing the result:
Extracting the latent factor in this manner is sometimes referred to as
extracting or estimating an indicator.
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In many cases, estimating \({\mathbf {g}}_t\), or any other rotation of \({\mathbf {f}}_t\), is as good as estimating \({\mathbf {f}}_t\) itself (see Sect. In the subroutine, we use smoothed states (see Sect. For each set of dimensions \(\{N,T\}\), the parameters \(\beta _i\) and \(\lambda _i\) are drawn 50 times. 5, and applied in an empirical example in Sect. The third equation is a first-order autoregressive equation for m, the variable capturing discretionary policy that affects interest rates. The paper has benefitted from discussions with several colleagues at the Ministry of Finance.
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The method we set up in this paper is a two-step procedure, in which parameters are first estimated by principal components, and then, given these estimates, the factors are re-estimated as latent states by the Kalman smoother. In practice, this is rarely the case. I call the latent factors affecting production e and the latent factors affecting interest rates m. Given these parameters, the minimum mean square error (minimum MSE) predictor of the static factors is the projection (see, e.
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Additionally, GDP is subject to substantial revisions as more source data become available. H. 4. The upper part of Table 2 displays the averages \({\bar{\varDelta }}_{T-s}^{{\mathcal {A}}4}\), and the lower part displays the ratios \({\bar{\varDelta }}_{T-s}^{{\mathcal {A}}4}/{\bar{\varDelta }}_{T-s}^{{\mathcal {A}}3}\) (a value below 1 indicates that \(\varOmega ^{{\mathcal {A}}4}\) is, on average, more accurate). g.
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Note that the variances of the errors \(z_t\) and \(u_{i,t}\) are scaled, so that the variances of \(f_t\) and \(\epsilon _{i,t}\) are 1 and \(\kappa _i\), respectively. The linear time series process \({\mathbf {x}}_t\) can be cast in the state space formwhere \(\varvec{\alpha }_t\) (\(k \times 1\)) is a latent state vector, \({\mathbf {H}}_t\) (\(N \times k\)) and \({\mathbf {T}}_t\) (\(k \times k\)) are possibly time-varying parameter matrices and \({\mathbf find out here (\(k \times q\); \(q\le k\)) is, in general, either the identity matrix or a selection matrix consisting of a subset of the columns of the identity matrix (see, e. 156).
dfactor also estimates the parameters of static-factor models,
seemingly unrelated regression (SUR) models, and vector find out here (VAR)
models by maximum likelihood.
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\[eq\_eq5\] \[def\_pv\] In the following we make use of [“PV”]{}: n0=\[hat\]. A recent demonstration is found in van den Bossche (2011). (independence) The idiosyncratic errors \({\mathbf {u}}_t = (u_{i,t},u_{2,t},\ldots ,u_{N,t})’\) and common shocks \({\mathbf {z}}_t\) are mutually independent groups: \(E({\mathbf {u}}_t{\mathbf {z}}_s’) = {\mathbf {0}}\) for all t, s. .